问题 解答题
已知椭圆E的中心在坐标原点O,经过两点A(1,
2
5
5
),B(-2,
5
5
).
圆C以点(2,0)为圆心,椭圆的短半袖长为半径.
(1)求椭圆E的标准方程;
(2)若点P是圆C上的一个动点,求
CP
OP
的取值范围.
答案

(1)设椭圆E的标准方程为mx2+ny2=1(m>0,n>0,且m≠n).

因为A(1,

2
5
5
),B(-2,
5
5
)在椭圆E上,所以
m+
4
5
n=1
4m+
1
5
n=1

解得m=

1
5
,n=1,满足条件

所以所求椭圆E的标准方程为

x2
5
+y2=1.

(2)由(1)知椭圆的短半轴长为1,所以圆心坐标为(2,0),半径r=1,

故圆C的方程为(x-2)2+y2=1.

设P(x,y),则

CP
=(x-2,y),
OP
=(x,y),

所以

CP
OP
=x(x-2)+y2=x2+y2-2x=2x-3.

因为(x-2)2+y2=1,所以(x-2)2≤1,即-1≤x-2≤1,得1≤x≤3.

所以-1≤2x-3≤3,即

CP
OP
的取值范围为[-1,3].

完形填空
When Mr. Hunt woke up in the morning, he felt his bedroom was very bright. He thought he ____ to turn off the light again. He sat up and tried to do that. And now he ____ it was the sunlight.
Mr. Hunt went to see their daughter in the city two days ago. She cooked some food for him ____ he left. But Mr. Hunt hoped to have breakfast in a restaurant near the park. And then he could walk ____ the lake. He put on his clothes and went out. The sun was shining brightly in the sky and the ____ was fresh (清新的) . But people were busy and he could see only some old people and children. He went in the restaurant and sat down at a table, waiting for the ____.
“Hi, Peter!” someone called him and he saw it was his ____ Richard. “Hi, Richard. Glad to meet you here.” “Glad to meet you, too,” said Richard. “I’m sorry , but I ____ my wallet at home . So could you lend fifty dollars to me, please?” “Sure, but I have only twenty five dollars in my ___.” “It doesn’t matter ,” said Richard , “ then you’ll owe(欠) me twenty-five dollars and I’ll owe you ____ dollars , too . That’ll be fair (公平的) .”
小题1:
A.knewB.hopedC.forgotD.liked
小题2:
A.guessedB.foundC.said D.told
小题3:
A.beforeB.afterC.whenD.because
小题4:
A.inB.on C.underD.around
小题5:
A.fruit B.cloudC.airD.water
小题6:
A.moneyB.glassC.answerD.menu
小题7:
A.friendB.workerC.driverD.waiter
小题8:
A.tookB.putC.forgotD.left
小题9:
A.shoesB.pocketC.shirtD.hand
小题10:
A.fifteenB.twentyC.twenty-five D.thirty
解答题